Numerical Analysis

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Numerical Analysis

and

Analytical Approximations

of

Slow-Roll Inflation Models

Master Thesis for the Physics track

(Quantum Universe specialization) at the University of Groningen

by

Ricardo de Ruiter s1690930

under supervision of Dr. Diederik Roest

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List of Figures

2.1 Horizon problem . . . 12

2.2 Horizon problem - solution . . . 16

2.3 Scalar power spectum Planck collaboration . . . 23

2.4 Chaotic inflation potential steepness . . . 24

2.5 Starobinsky inflation potential . . . 27

2.6 Cosmic microwave background . . . 29

2.7 Exclusion plot Planck collaboration . . . 36

2.8 Exclusion plot BICEP2 collaboration . . . 38

3.1 Attractor behavior of chaotic inflation models (normal r) . . . 42

3.2 Attractor behavior of chaotic inflation models (ln r) . . . 42

3.3 Chaotic inflation ξ-attractor (numerical): n_s, ξ . . . 46

3.4 Chaotic inflation ξ-attractor (numerical): r, ξ . . . 46

3.5 Chaotic inflation ξ-attractor (numerical): r, n_s . . . 47

3.6 Chaotic inflation ξ-attractor (numerical): ¹⁰log r, n_s . . . 47

3.7 Effect of ξ on CI potential . . . 48

3.8 Chaotic inflation: relative errors in n_s, neglecting φ_end . . . 51

3.9 Chaotic inflation: relative errors in r, neglecting φ_end . . . 51

3.10 Chaotic inflation: relative errors in ns, neglecting logarithms . . . 52

3.11 Chaotic inflation: relative errors in r, neglecting logarithms . . . 52

3.12 Chaotic inflation: relative errors in n_s, neglecting φ_end and logarithms . 53 3.13 Chaotic inflation: relative errors in r, neglecting φend and logarithms . 53 3.14 Parametric plot of r, n_s for wrong approximation . . . 55

3.15 Analytic expressions for the ξ-attractor: r, n_s . . . 57

3.16 Analytic expressions for the ξ-attractor: ¹⁰log r, ns . . . 57

4.1 Induced inflation ξ-attractor (numerical): n_s, ξ . . . 62

4.2 Induced inflation ξ-attractor (numerical): r, ξ . . . 62

4.3 Induced inflation ξ-attractor (numerical): r, ns . . . 63

4.4 Induced inflation ξ-attractor (numerical): ¹⁰log r, n_s . . . 63 5

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4.5 Induced inflation: relative errors in n_s, neglecting logarithms . . . 65 4.6 Induced inflation: relative errors in r, neglecting logarithms . . . 65 4.7 Induced inflation: relative errors in n_s, neglecting φ_end and logarithms . 66 4.8 Induced inflation: relative errors in r, neglecting φ_end and logarithms . 66 5.1 Close-up ξ → ∞ regime . . . 71

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List of Tables

C.1 Relative errors in n_s neglecting φ_end solving for φ_N . . . 98

C.2 Relative errors in r neglecting φ_end solving for φ_N . . . 98

C.3 Relative errors in ns neglecting logarithms solving for φN . . . 99

C.4 Relative errors in r neglecting logarithms solving for φ_N . . . 99

C.5 Relative errors in n_s neglecting φ_end and logarithms solving for φ_N . . . 100

C.6 Relative errors in r neglecting φend and logarithms solving for φN . . . 100

7

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Introduction 1

As the last project before graduation, we have taken up a research project in theoretical physics, specifically in the field of inflation. This research has been done at the Theoretical High-Energy Physics (THEP), a research group that is part of the Faculty of Mathematics and Natural Sciences of the University of Groningen.

Under supervision of Diederik Roest (whom we owe our gratitude for his careful and enthusiastic supervision), the head of the String Cosmology subgroup of the center of THEP, we have investigated the behavior of various models of inflation, specifically their recently discovered attractive behavior.

The aim of this research is to numerically investigate this behavior and find analytic approximations of the evolution of these models as the non-minimal coupling between the scalar inflationary field and gravity increases. The models we have investigated are mostly chaotic inflation [49] and induced inflation [28].

Analysis of both numerical and analytical cases has been done using Mathematica.

In chapter 2, an brief overview of the history around the development of inflation will be provided. This will also involve problems inflation solved in cosmology back in the days and the development of slow-roll inflation. In this discussion the results of large experiments of WMAP and Planck will also be touched upon, as well as the recent controversial measurements of the Antarctic BICEP2 experiment.

After that, in chapters 3and 4, the method and results of the numerical and analytic analysis of non-minimally coupled inflation models (the aforementioned chaotic and induced models) will be presented. This chapter will refer to the appendices B through D for detailed explanation of the working of the written programs and numerical data.

Then in the last chapter, chapter 5, we will discuss our methods and results and will try to find points to improve upon.

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Inflation 2

Why inflation? 2.1

In the earlier part of the twentieth century the field of cosmology made its appearance in physics. After the groundbreaking work of Einstein with his theory of general relativity and the later models of the universe by Friedmann (in particular his cosmological equations in the FLRW-metric), cosmology found itself describing the universe at large: a vast volume that could expand or contract and that could have some amount of curvature.

Independent from Friedmann, the Belgian priest Lemaˆıtre also derived the equations of the FLRW-universe. From their form and behavior he proposed that the could have been a moment in the far history of the universe, where all the universe was at this moment, originated from: the primeval atom, later dubbed the big bang. It was only a few years later that Lemaˆıtres theory found observational suggestions hinting at this particular scenario being correct. Hubble’s research showed that galaxies seemed to recede from our own, depending on their distance:

the more distant, the larger their velocity of recession.

The theory of an evolving universe strongly competed with the steady state universe theory that supposed the universe, over time, proved to maintain a certain status quo. In 1965 however, a discovery made by astronomers Penzias and Wilson showed that there was indeed an echo of the ‘primeval atom’ Lemaˆıtre had thought of. Their observation was a microwave background, the relic of a big bang, which had been predicted by earlier theoretical research.

Over the years many research had been done in the field of this cosmic microwave background (CMB), but this baby card of the universe did provide a problem that the standard big bang theory couldn’t solve: the radiation was uniform, with minute deviations, over all of the sky. This problem became known as the horizon problem.

Besides this problem, there were two other problems with the big bang scenario.

The first of these, proposed by physicist Dicke in the late 1960s, was known as the

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flatness problem. The second one was called the monopole problem.

In the next few sections we will describe the three problems and their implica- tions, followed by a section on how inflation, as proposed by Guth in 1980, solves each of these problems.

2.1.1 Horizon problem

The first relevant problem to describe is the horizon problem. As stated, the CMB appeared to be of a uniform temperature over the entire sky, but according to special relativity, it would not have been possible for all sky-regions to have been in causal contact with one another. Since the universe, as it had a beginning in the Big Bang, has a finite age, light has only had a finite amount of time to traverse it. CMB-photons we detect now come from the edge of our observable universe, and thus could only have had time to interact with a very small patch of sky indeed, which can be shown from calculation of the proper distances in the FLRW-background.

Even conceptually it can become clear, considering two CMB-photons coming in from opposite directions in the sky. As they are observed, they’ve only just reached our detectors, but show they are in near-perfect equilibrium, despite causal connection between them being impossible. It can be easily seen that this poses a large problem in big bang cosmology.

Figure 2.1: A schematic representation of the horizon problem, where the has been no causal contact between individual patches of sky [9].

The following compacted calculation from [9] will illustrate this. In it we

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2.1. Why inflation? 13

consider a flat FLRW metric, where instead of normal time we use the scale- invariant conformal time: dτ = dt/a(t).

In this metric photons will of course travel along null geodesics, meaning r(τ ) = ±τ + c with c some constant. The maximal distance traveled then becomes,

∆rmax(t) = lim

ti→0

Z t

ti

dt⁰ a(t⁰)

= Z t

0

dt⁰ a(t⁰)

= τ (t) − τ (0) (2.1)

From this so-called ‘co-moving horizon’ we find that τ =R (aH)⁻¹d ln a. If the universe if dominated by a fluid with equation of state w = p/ρ, then (aH)⁻¹ ∝

a¹²^(1+3w), which leads to τ (0) = 0 and:

∆r_max(t) ∝ a(t)¹²^(1+3w) (2.2) Here it is assumed w > −¹₃.

If we now define the angular region of the co-moving horizon at the moment the CMB was emitted, as θhor = dhor/dCM B, and rewrite our equations2.1 and2.2 for the co-moving horizon to something as a function of redshifts, then

τ₂− τ₁ = Z z2

z1

dz

H(z) =D(z1, z₂) (2.3)

leads to the relation

θ_hor = D(zCM B, ∞)

D(0, zCM B) (2.4)

with H(z) = H₀ q

Ω_m(1 + z)³ + Ω_γ(1 + z)⁴+ Ω_Λ (2.5) If we then fill in all the known parameters for equations 2.4 and 2.5 (i.e.

Ω_m = 0.3, Ω_Λ= 1 − Ω_m, Ω_γ = Ω_m/(1 + z_eq), z_eq = 3400 and z_{CM B} = 1100) we find that θ = 1.16°, hence:

θ_causal = 2θ_hor

≈ 2.3° (2.6)

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2.1.2 Flatness problem

Another problem that arises in cosmology is the flatness problem. From the Friedmann equation,

H² = 8πG 3 ρ − k

a² (2.7)

it can be seen that for a flat universe, where k = 0, we find a critical density describing the energy content of specifically the flat universe:

ρ_c= 3H²

8πG (2.8)

Defining a density parameter Ω = ρ/ρ_c, we can manipulate the Friedmann equation to the following form:

H² = 8πG

3 ρ_cΩ − κ a²

= H²Ω −κ² a² And hence:

(Ω − 1) = κ a²H² or |Ω − 1| = |κ|

a²H² (2.9)

The problem now comes from the factor a²H² in de denominator on the right- hand side of equation 2.9. In different cosmological models it can be found that this product provides an indication of scaling in the universe with time. In any model that had a radiation or matter dominated era (which together give a fair representation of the universe’s evolution up to about its current age), the scaling of a²H² is proportional to some t^−α with α > 0. This means that |Ω − 1| will grow proportional to t^α. Any small deviation from flatness (i.e |Ω − 1| = 0) will thus be magnified over the course of time. This is particularly troubling since our universe is observed to be approximately flat, with |Ω − 1| < 0.01. In order for this value to be this value in the present days, the value in the early years of the universe had to be tens of orders of magnitudes smaller, meaning the universe had to be extraordinarily flat as it came into existence.

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2.2. Field theoretical description 15

2.1.3 Monopole problem

Another problem to have come up in those days arose from the attempts to unify the four forces of nature. In those attempts, of which [27] is an early example, the unified theories always seem to predict the existence of primordial relic particles, magnetic monopoles being a particular type. From simple physical reasoning one could already imagine such particles to present: at energy scales of about 100 GeV we already observe the emergence of the W^± and Z⁰ as force carriers of a unified electroweak force. It isn’t hard to imagine additional unification leading to other new particles at even higher scales (such unification is at least expected to happen at the Planck scale, if not before that).

As inferred before, however, there’s a problem in having these particles being predicted by the models: they should be produced in vast amounts, such vast amounts (see [60,55], we should definitely have observed them already. Given that the search for such particles has long been ongoing, it’s suffice to say we would have noticed some papers on the topic should these particles have been detected.

2.1.4 Solving the problems

In a later section (2.3) we will present more about the way inflation works, but for now it’s suffice to say that inflation solves all three aforementioned problems.

The horizon problem is solved by the exponential expansion of the universe in it’s early stages. We recall the conformal time mentioned in section 2.1.1, and can write that, during inflation, the evolution of the scale factor can be written as a(τ ) = −1/Hτ . Assuming H ≈ constant, we find that at the Big Bang (a = 0), we have a time τ = −∞. We extend the time axis shown in figure 2.1 to times before τ = 0. Since the relation for exponential expansion breaks down at τ = 0, we will now speak of the end of inflation at that time, instead of the Big Bang [9].

Furthermore, recall equation 2.9. If we have an exponential expansion, we see that the denominator on the right-hand side of that equation becomes very, very small (considering a² is usually considerd to be (e⁶⁰)²). Instead of any deviations from flatness increasing over time, during inflation they are damped to negligible values. Last but not least, a period in which the universe expands by a factor e⁶⁰ extremely dilutes any high density of magnetic monopoles.

2.2 Field theoretical description

A field theoretical description of inflation is often given in terms of the Lagrangian formalism for fields. It’s suffice to say we are not going to present the complete derivation of this formalism, but we will briefly sketch the origin and use of it. For a more in-depth derivation, refer to [7,16, 23].

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Figure 2.2: A schematic representation of the solution to horizon problem, extending conformal time to the infinite past. Regions in the plot now had causal contact, solving the horizon problem [9].

The formalism stems from the Lagrangian description of a discrete system, say a chain or lattice of particles connected by ‘springs’. Taking the distance l between these particles in the limit l → 0 whilst the number of particles N → ∞, it is possible to make a transition from just describing the particle by it’s position x to a field, not only dependent on position, but time as well: φ = φ(~x, t).

It is easily recognized that the original equations of motion of the spring system become a new set of equations dependent on the field. We can, as we could in the original case, define a Lagrangian to describe the system:

L_i = T (φ_i) − V (φ_i) (2.10) It’s possible to forget about the whole lattice and just think of the field as something in space, or rather space-time (what we are more concerned about).

If it would then follow a specific trajectory through space-time, were we would generally use the principle of least action, depending on the defined field and it’s

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2.3. Slow-roll inflation 17

derivatives. Since we consider space-time now, we redefine the field as a function of space-time coordinates, φ(~x, t) → φ(x), where we implicitly assume x to be representing a position in any dimension:

S = Z

dtL(φ(x), ∂µφ(x)) or (2.11)

= Z

d^DxL (φ(x), ∂µφ(x)) (2.12) Here L is the Lagrangian density, defined as L = R dxⁱL . It can be proven that the action will be invariant under variation of the field and its derivatives within the Lagrangian density.

We can then define a scalar field, for which we take the following action (assuming we live in a Lorentz-invariant world of four space-time dimensions):

S = Z

d⁴x(−1

2∂_µφ∂^µφ − V (φ)) (2.13) (2.14) Here we have a kinetic term for the scalar field and a possible potential. This potential becomes a crucial term when we consider inflation, since it’s the shape of the scalar field potential that determines the evolution of the universe during the inflationary period.

The last modification we make to this action is by substituting it as a matter Lagrangian into the Einstein-Hilbert action, in which our scalar field has the possibility to interact with the space-time background:

S = Z

d⁴x√

−g(1 2R − 1

2∂µφ∂^µφ − V (φ)) (2.15) (2.16)

2.3 Slow-roll inflation

The inflation model as proposed by Guth was successful, but unfortunately had some problems of its own. In the years after he published his model, one of the first people to solve these problems was Andrei Linde [49]. His new model op inflation was based on the premise of a scalar field rolling down a potential hill, instead of the vacuum state tunneling Guth proposed. The fact that the rolling of the scalar was required to occur slowly made that this new type of

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inflation became known as slow-roll inflation. In the following subsections we will derive the slow-roll parameters and the corresponding observables. We referred to [45, 67,47, 58, 46,48] for additional information on these derivations.

2.3.1 Derivation of slow-roll parameters

To derive the slow-roll parameters we start with a FLRW metric:

ds² = dt²− a(t)²

dr²

1 − kr² + r² dθ²+ sin²θdφ²

(2.17) Here k is used to determine the type of curvature the universe has. From this we can derive the Friedmann equation:

˙a²

a² = 8πG 3 ρ − k

a² (2.18)

If we use the fluid equation from thermodynamics:

˙

ρ = −3˙a

a(ρ + p) (2.19)

perform some differentiation on equation 2.18 and substitute them into one another, we will derive the acceleration equation:

¨ a a − ˙a²

a² = −4πG(ρ + p) − k

a² (2.20)

For the following derivation we will only consider the case k = 0, since observations indicate we live in a flat universe. We also note that H = ˙a/a, hence the left-hand side of equation 2.20 becomes ˙H and hence:

3H² = 8πGρ (2.21)

H = −4πG(ρ + p)˙ (2.22)

Besides this, it is possible to derive the following relations for the stress-energy tensor for the scalar:

T^µν = δL

δ(∂_µφ)∂_νφ − η^µνL

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= ∂^µφ∂^νφ − 1

2η^µν[∂^ρφ∂_ρφ − V (φ)] (2.23) T⁰⁰ = ρ = 1

2

φ˙²+ V (φ) (2.24)

T_iⁱ = p = 1 2

φ˙² − V (φ) (2.25)

One of the assumptions here is that the field φ behaves as a perfect fluid and is isotropic in all directions, hence the spatial derivatives that appear in T^µν disappear. Also, we write η as the metric tensor, since we consider k = 0.

Then, using the relations 2.24 and 2.25 in equations 2.21 and 2.22, we obtain at the following relations:

3H² = 8πG 1 2

φ˙²+ V (φ)

(2.26)

H = −4πG ˙˙ φ² (2.27)

And from the fluid equation in 2.19we can also deduce the following, by filling in the T⁰⁰ and T_iⁱ results:

φ +¨ dV (φ)

dφ = −3H ˙φ (2.28)

We observe there is a type of friction term in this equation of motion (the φ-term), that provides us with a means to let our field roll down the potential.˙ If now we require that ˙φ² V (φ), i.e., the field rolls down slowly, the rewritten Friedmann equation2.26tells us we have a nearly constant amount of cosmological expansion, which could be useful if we try to inflate our universe. This requirement would also mean that the value of ˙H in equation 2.27 is small.

If we would now rewrite equation 2.20using the Friedmann equation from2.18, we find:

¨ a

a = −4πG

3 (ρ + 3p) (2.29)

H + H˙ ² = −−8πG

3 ˙φ²− V (φ)

(2.30) From this we observe, in order to have inflation, we require that p < ρ/3. This statement is equivalent to our previous statement: ˙φ² V (φ), and from this we also derive that the derivative of this approximation leads us to state | ¨φ| V_φ, where the subscript φ denotes a derivative with respect to φ.

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From 2.29 and 2.30 we can now derive a useful parameter:

H + H˙ ² = H² 1 + H˙ H²

!

= H²(1 − _H) (2.31)

_H = −H˙

H² (2.32)

If we want to rewrite this parameter in terms of the potential, using the slow-roll approximations, we write:

H² = 8πG

3 V (φ) (2.33)

3H ˙φ = V_φ (2.34)

Using 2.27, we find:

φ˙² = V_φ² 3H² H = −4πG˙ V_φ²

9H² H˙

H² = −4πGV_φ² 9H⁴

= − V_φ² 16πGV²

_V = 1 16πG

V_φ V

2

(2.35) This is the first slow-roll parameter. From equation 2.31 we can see that it should obey 0 < < 1 in order for inflation to occur. Remembering one of our slow-roll requirements, we put an additional constraint on : 1. Furthermore, from now on we will write 8πG = 1/M_{P l}² = 1.

Given that H is approximately constant under the slow-roll requirements, we find that we deal with a (near-)exponential expansion of the universe, hence a ∝ e^Hdt = e^−N, with N the number of so-called e-folds of inflation. We can then define:

dN = −d ln a = −Hdt (2.36)

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N = Z af

ai

d ln a = Z tf

ti

H(t)dt (2.37)

Note that the minus sign in front of N is absorbed in the reversal of integration limits, since N = 0 at the end of inflation (we integrate ‘into the past’). We can now rewrite Hdt:

Hdt = Hdt dφdφ

= H φ˙ dφ

= −3H²dφ Vφ

= −V V_φdφ N = −

Z φf

φi

V Vφ

dφ (2.38)

We used equations 2.33 and 2.34 in rewriting the first relation.

The next step is to also define a second dimensionless slow-roll parameter, which we will dub η. We will use the requirement that also the acceleration of the scalar field should be small: | ¨φ| |3H ˙φ| ≈ |Vφ|.

If this requirement holds, we find 3H ˙φ + V_φ ' 0, which we will manipulate a bit:

d

dt(3H ˙φ) ' d

dt(−V_φ) 3 ˙H ˙φ + 3H ¨φ ≈ −Vφφφ˙

3 ¨φ

H ˙φ ≈ −V_φφ

H² − 3 ˙H

H² (2.39)

Since we require | ¨φ| |3H ˙φ|, the left-hand side of equation 2.39is small. We already know that the second term on the right-hand side, equal to _H, is small as well. Hence, the term V_φφ/H² must be small too.

Here we then define η_H and η_V:

ηH = − φ¨

H ˙φ = 1 2

H¨ H ˙H

!

(2.40)

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η_V = V_φφ

V (2.41)

We thus find that, in order to have ‘successful’ inflation, 1 and |η| 1.

It is possible to define higher order parameters, but for this thesis, we will not consider them. As a last remark, when neglecting these higher order parameters, note that _V ≈ _H and η_V ≈ _H + η_H.

2.3.2 Observables

With the parameters derived in the previous section, it should be possible to define some observable quantities. We will go into their physical nature later (in section 2.5), but for now we will just make the assumption that the observables look like this.

The first observable is the spectral index of the scalar spectrum, n_s (often used as n_s− 1 as well). The second is the tensor-to-scalar ratio, r:

n_s = 1 + 2η − 6 (2.42)

r = 16 (2.43)

The first of these, the spectral index, is a measure of deviation from scale invariance. In the CMB, one can detect a power spectrum of the temperature fluctuations, as caused by the scalar field (as can be seen in figure 2.3).

If ns = 1 (or ns− 1 = 0), the power spectrum of the scalars is considered to be scale-invariant, which means that it’s power is independent on whatever the magnitude of the angular scale ` is. Any spectrum where n_s 6= 1 is called titled (which is why the spectral index is sometimes also called the spectral tilt). If then, for example, n_s < 1, we call the spectrum a red spectrum. This implies that there is more power at large scales (smaller `) than there is supposed to be for a scale-invariant universe. The reverse argument holds for ns > 1, a blue spectrum.

Here, there is less power at small scales instead.

Inflationary cosmologists and physicists will occasionally talk about the running of the spectral index as well, called αs for the scalar spectrum. In this thesis, we will not consider α_s since it’s practically irrelevant for the models considered here.

Other inflationary models do predict large running of the index however, so the parameter in itself has relevance to be observed, in order to ‘prove’ a specific model of inflation is a correct or valid one.

The second observable, r, concerns the amount of tensor perturbations with respect to scalar perturbations. These are the perturbations of the quantum fields in the primordial universe. The scalar perturbations form the basis for the temperature fluctuations we now observe in the CMB (in that sense, they are

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2.4. Toy models 23

Figure 2.3: Planck collaboration scalar power spectrum plot over the angular scales in the CMB. [20]

the seeds of structure formations, since without these perturbations the universe would have been completely uniform in temperature). The tensor perturbations are fluctuations arising from the predicted, yet still elusive, graviton. They form primordial gravitational waves, leaving a distinct pattern in polarization of the CMB.

A third observable, that’s less relevant for this research as well, is the tensor spectral index, denoted by n_t. It basically is a tensor-mode equivalent of the scalar spectral index.

2.4 Toy models

Now we have created our framework of slow-roll inflation, we will investigate the inflationary properties of the m²φ² chaotic inflation model and Starobinsky’s R² model to see what they predict for the slow-roll observables.

2.4.1 Chaotic inflation: m²φ²

We stated before that we require a slowly rolling scalar field in order to have successful inflation. There are many models in the inflation zoo that gives various

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predictions concerning the spectral indices and the tensor-to-scalar ratio. One of these models is chaotic inflation, a model which has a polynomial potential φⁿ. One may remark that such potential, at higher values of φ becomes quite steep (refer to figure 2.4. Could slow-roll inflation still be viable then?

Figure 2.4: Comparison of the relative steepness of chaotic inflation poten- tials for 0 ≤ φ ≤ 1.2, using n ∈ {²/3, 1,³/2, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Unsurprisingly, the answer to this question is yes (or else the model would not have been a viable one). We will show this in the following derivation. We take the n = 2 chaotic inflation model, where we have the following Lagrangian:

L =√

−g 1 2R − 1

2(∂φ)² − V (φ)

(2.44) V (φ) =1

2m²φ² (2.45)

Given our definitions of _V and η_V, as given by equations 2.35 and 2.41, we derive the following forms of the slow-roll parameters (leaving out the m² for convenience):

= η = 2

φ² (2.46)

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2.4. Toy models 25

Slow-roll ends whenever = 1, hence we find that φ_end =√

2. With this, we solve the e-folds equation from equation 2.38:

N = − Z φf

φi

V V_φdφ

= − Z φf

φi

φ 2dφ

=φ²_N

4 − φ²_end 4 φ_N =√

2 + 4N (2.47)

≈2√

N (2.48)

Where in the last relation we used that N is quite large. This expression for φ_N can be filled in into the definition of n_s and r, which we presented in equations 2.42 and 2.43:

n_s =1 − 8

2 + 4N (2.49)

≈1 − 2

N (2.50)

r = 32

2 + 4N (2.51)

≈8

N (2.52)

Now it is commonly found that inflation had about 50 to 60 e-folds. As such we will use a value within this range. Since we will mostly hold on to 60 e-folds of inflation throughout this thesis, we will present the values of n_s and r for N = 60 (with the values of the parameter within parentheses corresponding to the large-N

approximations given in equations 2.50 and 2.52):

n_s≈0.9669 (0.9667) (2.53)

r ≈0.1322 (0.1333) (2.54)

We find that this particular chaotic inflation model predicts that the universe is not scale-invariant, but should have a red-tilted spectrum. We also find that a fair amount of the power spectrum should be made up by tensor perturbations.

Furthermore, we will refer to these values in the next part of this chapter, in one of the subsections in section 2.6, specifically figure 2.8 (where both this model and the model we are about to discuss next will be featured).

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2.4.2 Starobinsky inflation: R²

The second model we will look into is the R² Starobinsky model. This model contains a curvature-squared term in the Lagrangian in addition to the Einstein- Hilbert term (neglecting for a moment mass-terms like m and M_{P l}):

L =√

−g 1

2R − 1 12R²

(2.55) The R² term in the Lagrangian can be rewritten [3] in a linear representation:

L =√

−g 1

2R − Rψ − 3ψ²

(2.56) We can now rewrite this equation, which has a direct interaction between the curvature and the field ψ (which we call the Jordan frame), to another frame where we can interpret the model: the Einstein frame. In that frame we will not have direct interaction of the curvature and the field. We do this by means of a frame function Ω which determines the transformation of the metric g_µν → Ω⁻¹g_µν.

It turns out, that if we take a frame function Ω = 1 + 2ψ/M , we arrive at this Einstein-frame Lagrangian, which can be written as a scalar field version of the Starobinsky model:

L =√

−g 1 2R − 1

2∂_µφ∂^µφ − 3

4(1 − e⁻

√2 3φ)²

(2.57) We observe that the last term in equation 2.57is inflation’s potential (see figure 2.5). Before we continue with any calculation, we would like to note the following:

inflation happens for large values of the field, with φ > M_{P l}. We note that φ is taken to be relative to the Planck mass, and the, if φ 1, we find e^−αφ 1 and e^−2αφ e^−αφ.

We then calculate and η:

=1 2



 2

q2

3e⁻

√2

3φ− 2q

2

3e⁻²

√2 3φ

1 − 2e⁻

√2

3φ+ e⁻²

√2 3φ





2

≈4 3e⁻²

√2

3φ (2.58)

η =−⁴₃e⁻

√2

3φ+ ¹⁶₃e⁻²

√2 3φ

1 − 2e⁻

√2

3φ+ e⁻²

√2 3φ

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2.4. Toy models 27

Figure 2.5: Plot of the scalar potential for Starobinsky inflation for field values −1 ≤ φ ≤ 10.

≈ − 4 3e⁻

√2

3φ (2.59)

We also take a differential form of the e-folds equation 2.38 and find the following:

dφ dN =V

Vφ

=2 r2

3e⁻

√2

3φ(N ) (2.60)

dN =1 2

r3 2e

√2

3φ(N )dφ

N =3 4e

√2

3φ(N )

(2.61) With this relation we identify that:

= 3

4N² (2.62)

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η = − 1

N (2.63)

As such, we find:

n_s =1 − 2 N − 9

2N² (2.64)

≈0.9654 (2.65)

r =12

N² (2.66)

≈0.0033 (2.67)

We see that Starobinsky inflation provides a similar spectral index as φ² chaotic inflation (equal if neglecting the N² term in equation 2.64), but it has a much lower tensor-to-scalar ratio. We will refer to these values in section 2.6 as well.

2.5 Cosmological Perturbation Theory

We have now firmly established the slow-roll formalism of inflation and have understood how inflation solves the three main theoretical problems of the Big Bang model. However, inflation offers us much more than this.

With the recent observations of the CMB, the scientific community has seen the seeds of structure formation in the temperature fluctuations of the background radiation (figure 2.6).

However, the quantum nature of the scalar field that drives inflation, as well as the presumed-to-exist graviton, allows for the existence of the very scalar and tensor perturbations we observe to be frozen inside of the CMB.

This can be seen by doing cosmological perturbation theory on the near- homogeneous background of space-time, which we will call X(t, x) for a specific quantity X. The homogeneous background of this qunatity will be called ¯X(t), where we drop the x-dependence since we deal with a homogeneous background [9,64].

From these quantities it is possible to define linear perturbations:

δX(t, x) =X(t, x) − ¯X(t) (2.68)

At the moment of inflation, the two most important quantities are the scalar inflaton φ and the space-time metric g_µν. In the early universe, perturbations of these two quantities are important for the eventual further evolution of the universe. In this these we will not go into the full derivation as done in [9, 64,46].

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2.5. Cosmological Perturbation Theory 29

Figure 2.6: An image of the Cosmic Microwave Background made by the Planck satellite.

Instead, we will describe the methods by which we can van derive the important results these perturbations yield.

In general this derivation starts from perturbing a flat FLRW background, for both the scalar and tensor perturbations.

φ(t, x) = ¯φ(t) + δφ(t, x) (2.69) g_µν(t, x) =¯g_µν(t) + δg_µν(t, x) (2.70) Where a general line element in space-time is given by

ds² =gµνdx^µdx^ν

= − (1 + 2Φ)dt² + 2aBidxⁱdt + a²[(1 − 2Ψ)δ_ij +Eij] dxⁱdx^j (2.71) Bi =∂iB − Si

∂ⁱSi =0

Eij =2∂_ijE + 2∂(iFj)+ h_ij

∂ⁱFi =0

hⁱ_i =∂ⁱh_ij = 0

In this decomposition of the perturbed background, we find that S and F are vector modes, which are not present during inflation. However, since we

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will consider all quantities in the universe to be a perfect fluid, the B- and E -contributions will disappear as a whole, rather than part of them:

ds² = − (1 + 2Φ)dt²+ a²[(1 − 2Ψ)δ_ij+ h_ij] dxⁱdx^j (2.72) The tensor perturbations of the metric are gauge-invariant, but the scalars are not. As such, a certain coordinate gauge has to be chosen for further calculations, leading to forms of Φ, B, E and Ψ that leave the line element invariant.

Considering that most energy in the universe during inflation comes from the inflation, inflaton fluctuations (present in the stress-energy tensor) will influence the space-time geometry, as follows from the Einstein equations. As such, we have density and pressure fluctuations as well.

ρ(t, x) = ¯ρ(t) + δρ(t, x) (2.73) p(t, x) =¯p(t) + δp(t, x) (2.74) Leaving the exact nature of these gauge transformations to the reader, it is possible to define gauge-invariant scalar quantities ζ (so-called curvataure perturbation on uniform-density hypersurfaces) and R (comoving curvature perturbation).

One can show that these perturbations are equal on scales larger than the horizon and during slow-roll inflation as well. At scales larger than this horizon (k aH, when the wavenumber is smaller, and hence the wavelength is larger than the horizon) both quantities are conserved: frozen into the background.

Since both of these are equal, we can continue with the comoving curvature perturbationR. It is possible to create a two-point correlation function, ξ_R(r) = hR(x)R(x+r)i, using that ξ depends only on distance rather than the direction of the vector r due to the presumed isotropy of the universe.

Considering the Fourier transform of R, Rk = AR d³xR(x)e^−ik·x, we can create an ensemble average of a pair of those:

hRkRk’i =A² Z

d³xe^{−i(k+k’)x}

Z

d³rξ_R(r)e^−ik·r

Using δ(k) = BAR d³xe^±ik·x

hRkRk’i =A

Bδ(k + k’) Z

d³rξ_R(r)e^−ik·r

=1

BPR(k)δ(k + k’)

=(2π)³P_R(k)δ(k + k’)

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2.5. Cosmological Perturbation Theory 31

Here we define the power spectrum as Fourier transform of the correlation function:

PR(k) =A Z

d³rξ_R(r)e^−ik·r (2.75)

From this one can define the variance σ_R² = ξ_R(0) = BR d³kP_R(k) = R d ln k∆²_R(k), as such:

∆²_R(k) = k³

2π²PR(k) (2.76)

With a long derivation starting from canonically quantizing a field, depending on the Einstein-Hilbert Lagrangian in presence of a scalar field, rewritten in terms of the curvature fluctuation, it is possible to find the form of the power spectrum P and by the above, the variance of these perturbations. This formalism can be done for both the scalar perturbations and tensor ones that were present during the inflationary epoch. Without going into full detail of the derivation, we find that these equations for the variance are:

∆²_R(k) = ∆²_s(k) = 1 (2π)²

H² M_{P l}²

H²

φ˙² (2.77)

≈ 1 24π²

V M_{P l}⁴

1

_V (2.78)

2∆²_h(k) = ∆²_t(k) = 2 π²

H²

M_{P l}² (2.79)

≈ 2 3π²

V

M_{P l}⁴ (2.80)

First, from the Hubble definition, we can find that the H²/ ˙φ²-term on the right-hand side of equation 2.77reduces to 1/_H (using _H = −d ln H/dN ), and as such we can define the ratio of the variance (and hence power) of tensor to scalars to be:

r =∆²_t(k)

∆²_s(k)

=16H (2.81)

Aside from this parameter, it is also possible to define a quantity which measures how much scale dependence the power spectra show. Since we have seen that

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the wavenumber k is related to the scale of the perturbations, we can find that the derivative of the power spectra with respect to k yields a measure of this scale-dependence:

n_s− 1 =d ln ∆²_s

d ln k (2.82)

=d ln ∆²_s dN

dN d ln k n_t=d ln ∆²_t

d ln k (2.83)

=d ln ∆²_t dN

dN d ln k

Given our earlier relation, it is possible to rewrite these equations and find that, in the X_H-formalism,

n_s− 1 =2η_H − 4_H

n_t= − 2_H (2.84)

Using that _H ≈ _V and η_H ≈ η_V − _H, we arrive at our previous definition of n_s and r as given in equations2.42 and 2.43.

2.6 Experiments

Given a theory and its resulting observables, it is of course possible to design an experiment to detect the needed signals and analyze the data to obtain the true values of these observables. Several notable space-based experiments have been designed specifically for detecting the important cosmological parameters, including those relevant for inflation. All these measurements have been done on the CMB. The two defining experiments have been the American Wilkinson Microwave Anisotropy Probe, or WMAP, and the European Planck satellite.

2.6.1 Detecting inflation

Given the power spectra we found in section 2.5, the only thing left before we can untangle the nature of inflation in our universe, is finding a way to extract the necessary data from the CMB.

In the CMB all-sky map, as presented in figure 2.6, we see the seeds of structure formation: tiny anisotropies with respect to the background of 2.7K, the fluctuations from this value at most of the order 10⁻⁵K.

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2.6. Experiments 33

These fluctuations can be described [9] by a function ∆T (ˆn), with ˆn being the direction in the sky. From this fluctuation functions it is possible to construct a so-called harmonic expansion of the map, given by,

Θ(ˆn) = ∆T (ˆn) T₀ =X

`m

a`mY`m(ˆn) (2.85)

a_`m = Z

dΩY_`m^∗ (ˆn)Θ(ˆn) (2.86) We recognize the quantum mechanical spherical harmonics in the functions Y_`m. As one would expect, ` here refers to a certain `²-pole moment (` = {0, 1, 2, 3, . . .}:

monopole, dipole, quadrupole, octopole. . . ) and m = −`, . . . , = `. The functions a are then called the multi-pole moments.

From the last equation, 2.86, one can construct a correlation function of temperature fluctuations: C_`^{T T}. This correlation function in itself is an invariant angular power spectrum and useful in analyzing the data that comes from the CMB.

Since we deal with the temperature fluctuations, we have to consider the scalar perturbations, which were related to curvature perturbations R. In order to have a correspondence between these perturbations and the temperature fluctuations, we also define a transfer function ∆_{T `}(k), which can be approximated at the large scales we observe in the CMB:

a_`m =4π(−i)^`

Z d³k

(2π)³∆_{T `}(k)RkY_`m(ˆk)

`

X

m=−`

Y_`m(ˆk)Y_`m(ˆk⁰) =2` + 1

4π P_`(ˆk · ˆk⁰)

∆_{T `}(k) =1

3j_`(k [τ₀− τ_rec])

Note that in the transfer functions, the functions j are Bessel functions. These functions peak at the moment that k(τ₀− τ_rec) ≈ `. Then, given these functions, the correlation function becomes:

C_`^{T T} = 1 2` + 1

X

m

ha^∗_`ma_`mi

=2 π

Z

k²dkP_R(k)∆_{T `}(k)∆_{T `}(k)

= 2 9π

Z

k²dkP_R(k)j_`²(k [τ₀− τ_rec])

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∝k³PR(k)|_k≈`/(τ₀−τ_rec)

Z

d ln xj_`²(x)

`(` + 1)C_`^{T T} ∝∆²_s(k)|_k≈`/(τ₀−τ_rec) (2.87)

∝`ⁿ^s⁻¹ (2.88)

However, next to these temperature fluctuations, there are also polarizations within. The polarizations are caused by Thomson scattering of photons on electrons after recombination. It happens that this only occurs on photons fields which have a quadrupole component. This moment is obtained by the photon close to the moment of recombination and any anisotropies of temperature in the CMB will then also cause a slight anisotropy in polarization.

These polarizations require a slightly different way of analysis, and this is done by introducing a function of the multi-pole moment for a spin-2 particle, a±2,`m. This function can be written down in the form of two polarizations: E-mode and B-mode,

a_E,`m = −1

2(a_2,`m+ a−2,`m) (2.89)

a_B,`m = − 1

2i(a_2,`mia−2,`m) (2.90)

Much like the harmonic expansion of the temperature fluctuations (equations 2.85 and 2.86), we can write a similar function for the E- and B-modes:

E(ˆn) =X

`,m

a_E,`mY_`m(ˆn) (2.91)

B(ˆn) =X

`,m

a_B,`mY_`m(ˆn) (2.92)

The nature of these polarizations is as follows:

E-modes around cold spots are radial

E-modes around hot spots are tangential

B-modes around cold spots are counter-clockwise curls

B-modes around hot spots are clockwise curls

It is clear that, given the nature of perturbations we have investigated, scalar perturbations, which cause temperature fluctuations, are only able to produce E- mode polarizations (they create hot and cold spots). However, tensor perturbations, which produce gravitational waves that stretch space in one direction and compress

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2.6. Experiments 35

it in the perpendicular one, can enhance (or produce) E-modes, but can also create B-modes.

This point is significant, since B-modes are thus a direct hint of tensor perturbations in the primordial universe.

With the polarizations in mind, we can construct three additional correlation functions:

C_`^EE ≈(4π)² Z

k²dkPR(k)∆²_E`(k) (2.93) C_`^{T E} ≈(4π)²

Z

k²dkPR(k)∆_{T `}(k)∆_E`(k) (2.94) C_`^BB ≈(4π)²

Z

k²dkPh(k)∆²_B`(k) (2.95) As stated in [9], there are no correlation functions for TB and EB due to symmetry reasons. Besides this, we do note that there are three functions now (TT, TE and EE) which provide information on the scalar power spectrum, while

there is only one such function for the tensor modes.

As we have seen now, it is possible using these correlation functions to extract information on inflation from the CMB. This provides scientist with a way to test the theory, as has been done extensively by the following experiments we will discuss.

2.6.2 WMAP

The WMAP mission was the follow-up of the Cosmic Background Explorer (COBE) that launched in 1989. The experiment contained a broader range of mission goals than COBE and was specifically aimed to take more detailed measurements of the CMB.

Owing to its higher sensitivity, WMAP was able to distill more accurate readings of the cosmological parameters, and as such it was discovered the universe contained far more dark ‘stuff’ than actually detectable baryonic matter.

Of particular interest to this thesis was the fact that WMAP also made a measurement of the scalar spectral index and the tensor-to-scalar ratio. The earliest result of the first of these parameters was published a year after initiating data acquisition, putting the value at n_s = 0.99 ± 0.04 at 68% CL, meaning the universe could still be either scale variant or invariant.

After additional years of acquiring data [10], these numbers improved until, eventually, the final results of WMAP put the value for the spectral index to n_s = 0.972 ± 0.013 (at 68% CL) while the tensor-to-scalar ratio was bounded to be r < 0.38 (at 95% CL). If data from other experiments is included, then

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n_s = 0.9608 ± 0.0080 and r < 0.14 at the same levels of confidence. These values were not completely conclusive yet as in to say whether the universe is indeed scale invariant or not, and whether there should actually exist non-vanishing tensor perturbations.

2.6.3 Planck

The European counterpart of WMAP was Planck, a mission that launched in the spring of 2009, about eight years after WMAP. The premise of Planck was to deliver more sensitivity and hence a higher resolution.

In 2013, the first results of Planck were published, and they put more stringent bounds on the parameters of inflation. For the spectral index this means that n_s = 0.9616 ± 0.0094 (68% CL) and, including other experiments (of which WMAP results are also part), ns = 0.9608 ± 0.0054. For the tensor-to-scalar ratio the value improved to r < 0.11 at 95% CL, but this allowed for a non-existing amount of tensor perturbations, r = 0. The spectral index results however, indicate we live in a universe where there is a slight off-balance from complete scale invariance (figure 2.7).

Figure 2.7: Planck collaboration plot of 68% and 95% CL regions for ns

and r, compared to several predictions of theoretical models. Note the black and orange lines corresponding to our toy models of φ² and R² inflation.

Another thing that stands out from the results is that many at the very heart

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2.6. Experiments 37

of the allowed region of n_s, r published by Planck lies Starobinsky inflation, which thus seemed to be the most favorable model of inflation at that time.

2.6.4 BICEP2

All in all, the constraints on the allowed inflation models (those not yet excluded by experimental evidence) have tightened over the years, but the data of both these major experiments still allowed for a great deal of models with varying background (including models emerging from string theory or supergravity).

The next big step would be to put a stronger constraint on the tensor-to-scalar ratio. Many ground-based experiments are taking data on the polarization modes in the CMB, in which the primordial tensor perturbations would leave a distinct fingerprint in the form of B-mode polarization (the ‘curl’-like polarization pattern we mentioned in 2.6.1).

One of these experiments is the Antarctic-based experiment BICEP2, which takes readings from a small patch of sky in the so-called southern hole, a small area of the cosmic background that, according to dust models and maps, was fairly devoid of background noise.

Results of this experiment, which did observations from 2010 to 2012, were published in early 2014, announcing they had indeed detected these elusive B-modes in the CMB, leading to r = 0.20^+0.07_−0.05, with r = 0 excluded at 7σ, or 5.9σ after subtracting foreground [25].

The claim made by the collaboration was shocking as it has become controversial.

One of the first notable things that stood out from the combined data of BICEP2 and Planck was the fact that, in the sweet spot of their data, was the earliest model of slow-roll inflation, φ², while Planck’s highly-favored Starobinsky model seemed to become excluded instead (see figure 2.8)

The publication sparked a seeming tsunami of papers on the subject, ranging from papers backing up the claim from various theoretical viewpoints [31,19,71,8], papers (theoretically) expanding upon the data (for example by analysis of existing models) [18,17, 51, 32,52, 5, 59,70, 56, 68, 2], to papers finding ways to rewrite certain models to fit the newly found data [37, 69, 63, 22, 57, 30, 44], papers searching for alternative explanations [24,36], to papers criticizing the ‘discovery’

[61]. One of the claims of authors from the last group is the fact that the used dust models of the BICEP collaboration were too conservative, and that the actual dust in the region in which BICEP2 observed the CMB could contribute a lot more to the B-mode polarizations than assumed.

The Planck satellite is expected to publish all-sky dust maps in late 2014, and it is expected that their collaboration will release some data on polarization as well. However, the initial excitement of the fact that r 6= 0 has mostly subsided,

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Figure 2.8: Plot of 68% and 95% CL for n_s and r, combined from several experiments for certain models [53]. Note the black line for φ² inflation.

and the claim more dust is present in the southern hole has been backed up a lot over the months following the publication.

On top of waiting on Planck’s results, it could also prove important to wait for the results of BICEP2’s successor, the Keck array, which extends upon it’s predecessor by increasing the sensitivity and range of microwave bands observed.

To conclude, as to why this presumed discovery caused such an upheaval in the scientific community, we can state that by observing a finite value of r, we had a way to determine the actual size of the slow-roll parameters. Given the data published by Planck, r < 0.11, the value of could still have been anything of 0 < . 0.007. Having an exact range of values of not only allows to find an exact value of η, but also allows for finding the exact amount of scale-dependence of tensor modes (n_t), as well as allow us to pin-point the exact energy scale at which inflation occurred. In this light, having a set value of r provides scientists with a lot more insight into the deeper mechanisms of inflation: there is a reason

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2.6. Experiments 39

everyone would want to find the exact value of r.

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3

Chaotic inflation: ξ-attractor

3.1 Introduction

In recent studies by independent collaborations, it has been suggested there exist links between otherwise seemingly independent models of inflation, by introducing a non-minimal coupling between the gravity sector and the scalar field driving inflation. One collaboration [50] has shown that there exists an attractor point for non-minimally coupled models of chaotic inflation (CI). This point provides CI models with various powers of the potential with values of observables corresponding to those of Starobinsky inflation [65, 66] (section 2.4.2). As of now, these values are perfectly within the limits set on those observables (as seen in figure 2.7) by the Planck collaboration [21].

From the figure 3.1 it can be seen that, even though at minimal coupling, the different models of CI gives widely varying predictions for the spectral index and tensor-to-scalar ratio, as soon as coupling between the two sectors is turned on the predictions start to move into the allowed region of Planck, at first parallel, but afterwards with a spiraling attractive behavior (figure 3.2). This is the case for all CI models considered by [50] (and other models besides CI considered in the paper as well), despite them initially falling outside of the region.

For their approach, Kallosh, Linde and Roest used a frame function Ω (φ) = 1 + ξf (φ), with f (φ) = φ^n/2 (note how we get back our toy model from section 2.4.1 by setting ξ = 0). They investigated the behavior of the model in the small and large coupling regimes. Specifically for the small coupling regime they found indeed that the models, for all n, initially show a slope of −1/16 in the ns, r-plane, as could already be seen from figure 3.1. In their investigation of large coupling, after some approximations, they found that all chaotic inflation models ‘reduce’ to a Starobinsky model, thus yielding exactly the same results as that model in the limit ξ → ∞.

Another approach of non-minimal coupling was shown by Bezrukov and Gor- bunov [13], in which they studied the effect of a non-minimal coupling in the

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Figure 3.1: Attractive behavior of CI models for n = 2/3, 1, 2, 3, 4, 6, 8 in the ns, r−plane [50].

Figure 3.2: Attractive behavior of CI models for n = 2/3, 1, 2, 3, 4, 6, 8 in the ns, ln r−plane [50].

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3.2. Chaotic inflation: numerical approach 43

specific case of non-minimally coupled Higgs Inflation (HI). Their method used an approximation in solving for φ_N, the initial field value, as function of the number of e-folds of inflation N :

N = 3 4

ξ + 1 6

φ²_N − φ²_end− ln 1 + ξφ²_N 1 + ξφ²_end

(3.1) For the cases where ξ → 0 and ξ → ∞, they neglected the logarithmic term in the expression of N to find analytic expressions for φ_N: φ²_N → 8(N + 1) and ξφ²_N → (4N/3) in the respective cases. They then introduced an interpolation for the intermediate case where ξ has a finite value:

(1 + 6ξ)φ²_N = 8(N + 1) (3.2)

From the last equation it can be easily seen that for the limits ξ → 0 and ξ → ∞ reproduce the expressions for the field φ_N mentioned before. According to their calculations the error within the observables r and n_s where less than 10%, compared to the numerical solution to the problem.

Chaotic inflation: numerical approach 3.2

One of the first aims of this thesis was to reproduce the results of the paper by Kallosh, Linde and Roest [50]. In this, we considered the model of CI, with the final aim to try and find a suitable analytic approximation for the observables as a function of the number of e-folds N , the power of the field n (as derived from the potential φⁿ) and the non-minimal coupling ξ (something that will be discussed in section3.3. Doing the numerical analysis first provides a numerical background to which comparisons can be made when considering the analytical approximations we will investigate later on.

3.2.1 Method

The initial reproduction has been done using code in Mathematica. This code can be found in its completeness in the B (expressions of chaotic inflation slow-roll parameters and observables will be given inA.1, since for the purpose of explaining the code, for now only certain steps of the process will be discussed. The first input to the code are the formulas considered in [50]. The more straightforward of these are the Jordan frame potential V_J = f²(φ) = φⁿ, the frame function Ω(φ) = 1 + ξf (φ) and the Einstein frame potential V_E = V_J/Ω(φ)².